Develop an understanding of very large and very small numbers, estimation, mental calculations, and how to check if an answer is reasonable An introduction to generalising, and how to set out a verification or proof of a numerical generalisation to prepare for Algebra
Conceptual Framework
relationships quantity, generalisation, simplification, equivalence in the context of Identities and Relationships
Statements of Inquiry
Students will understand how different number types (N,Z,Q,R) arise in different contexts, and that fluent manipulation of these enables a more effective understanding of those contexts.
Factual
Exact or approximate - which is more useful?
What is 'approximately' equal?
When should we estimate?
How and why do we estimate?
How do we round?
Significant figures or decimal places?
Conceptual
Recurring decimals - Is \(0.\overline 9\) equal to one?
Why do different numbers terminate, recur, or not recur as decimals?
What are irrational numbers and when do they arise?
Debatable
The logic and power of indices?
From repeated multiplication to cube roots using indices
Can one example (or one million) prove a result?
Description
A number or quantity can often be expressed in many different ways, and with varying degrees of accuracy. In this unit, students will explore the different types of number and their links, such as prime numbers, rational and irrational numbers, repeating decimals, various types of indices, and the ways in which most of these naturally arise from practical problems. The unit includes the study of very large and very small numbers, how to represent and manipulate these, with and without a calculator, and practical examples of these. Students will learn through experience how measurement and related calculations inevitably lead to approximation, the consideration of an appropriate degree of accuracy, and the need to round answers to a suitable number of significant figures. At the same time, students will gain a greater appreciation for the importance of estimation and approximation, allied to mental Maths methods, and begin to develop a sense of when a numerical answer looks ‘about right’ so that they can critically evaluate answers and information. Through investigations students will learn to recognise number patterns, and also gain a greater understanding of how to generalise numerical results, and the difference between showing and verifying a result. They will also be introduced to the idea of a simple mathematical proof and how to present it.
Learning Outcomes
Using prime factor decomposition to find LCM, HCF efficiently
Represent rational numbers in a variety of forms including percents, fractions & decimals
Find, without a calculator, the following percentages of a number: 5%,10%,15%,90%,95% (17.5%)
Calculate the percentage increase and decreases of a quantity
Use the multiplier method for percentage increase and decrease
Use reverse percentages to find the original price for a sale item
Investigate patterns with fractions
Express fractions as terminating or repeating decimals
Convert any repeating decimal to a fraction by observing patterns (1/9,1/99 etc)
Identify fractions that give terminating, repeating, non-repeating decimals
Appreciate how irrational numbers and surds arise, and perform operations with them
Use the index laws including negative integer indices to simplify expressions involving exponents
Use index notation to express, simplify or compute positive, negative and zero powers of rational numbers (not algebraic)
Understand a fractional index such as ½ and ⅓ for roots and evaluate them on a calculator
Compute integer powers of positive or negative fractions and integers
Convert both large and small numbers to/from standard form
Write numbers in standard Index form with and without calculator
Perform operations on numbers in standard form
Round values to the correct number of significant figures and/or decimal places
Explain the implication(s) of giving 3,4,5 significant figures in an answer
Use estimation effectively to determine the reasonableness of an answer
Develop and use strategies to make reasonable estimates when finding sums, differences, products and quotients of rational numbers
Compare and order rational and irrational numbers
Compute the square roots and cube roots of numbers which are perfect squares and perfect cubes
Compute the square roots of large and small perfect squares involving multiples of powers of 10
Compare and order combinations of rational and irrational numbers, including \(\pi\), \(\phi\), and expressions involving these
Give examples of the following sets of real numbers: odd, even, prime, positive,negative, natural, integers, rational, irrational
Show how to set out a simple (numerical and algebraic) verification or proof